Average Error: 9.2 → 0.1
Time: 28.3s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{1 + x} \cdot \left(1 + \frac{x}{y}\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{1 + x} \cdot \left(1 + \frac{x}{y}\right)
double f(double x, double y) {
        double r28163857 = x;
        double r28163858 = y;
        double r28163859 = r28163857 / r28163858;
        double r28163860 = 1.0;
        double r28163861 = r28163859 + r28163860;
        double r28163862 = r28163857 * r28163861;
        double r28163863 = r28163857 + r28163860;
        double r28163864 = r28163862 / r28163863;
        return r28163864;
}
↓
double f(double x, double y) {
        double r28163865 = x;
        double r28163866 = 1.0;
        double r28163867 = r28163866 + r28163865;
        double r28163868 = r28163865 / r28163867;
        double r28163869 = y;
        double r28163870 = r28163865 / r28163869;
        double r28163871 = r28163866 + r28163870;
        double r28163872 = r28163868 * r28163871;
        return r28163872;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.2
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.2

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}}\]
  4. Using strategy rm

  5. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{x}{x + 1} \cdot \left(\frac{x}{y} + 1\right)}\]

  6. Final simplification0.1

    \[\leadsto \frac{x}{1 + x} \cdot \left(1 + \frac{x}{y}\right)\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))